X1653

5th EVANS PERSPECTOR

X1653 = X1081-aleph conjugate of X1653
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 7 }: Centroid–Gergonne Line { 1, 16 }: Central Line L(1, 16) { 46, 1276 }: Central Line L(46, 1276) { 395, 1081 }: Central Line L(395, 1081) { 1082, 1100 }: Central Line L(1082, 1100) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_7 = GERGONNE POINT (0.5903, 0.2769) X_1 = INCENTER (0.5714, 0.2333) X_16 = 2nd ISODYNAMIC POINT (1.8453, 2.3239) X_46 = -CEVA CONJUGATE OF X(1) (0.6917, 0.4543) X_1276 = 2nd EVANS PERSPECTOR (4.3665, 9.0809) X_395 = MIDPOINT OF X(14) AND X(16) (1.3794, 1.1048) X_1081 = POINT ALPHECCA (0.6056, 0.3226) X_1082 = POINT ALPHERATZ (0.5483, 0.1908) X_1100 = COMPLEMENT OF X(319) (0.5787, 0.2419) X_1653 = 5th EVANS PERSPECTOR (0.6584, 0.3759) A B C X2 X7 X1 X16 X46 X1276 X395 X1081 X1082 X1100 X1653 (a, b, c) = (5, 6, 7) | AB = 1

X1653 (5th EVANS PERSPECTOR) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \sqrt{3/4} + \sin(A - \pi /3) - \sin(B - \pi /3) - \sin(C - \pi /3)\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(-0.256214066828\)
▸ Alternate Coordinates3 forms

  • Trilinears: \(- \sqrt{3/4} + \cos(A + \pi /6) - \cos(B + \pi /6) - \cos(C + \pi /6)\)

  • Trilinears: \(\sqrt{3} \tan(A/2) - \tan(B/2) \tan(C/2) : \sqrt{3} \tan(B/2) - \tan(C/2) \tan(A/2) : \sqrt{3} \tan(C/2) - \tan(A/2) \tan(B/2)\)

  • Trilinears: \(\sqrt{3} \tan(A/2) - (b + c - a)/(a + b + c) : \sqrt{3} \tan(B/2) - (c + a - b)/(a + b + c) : \sqrt{3} \tan(C/2) - (a + b - c)/(a + b + c)\) (M. Iliev, 5/13/07)

▸ Incident Central Lines5 lines

{1, 16}, {2, 7}, {46, 1276}, {395, 1081}, {1082, 1100}

▸ Functional Relations2 relations